I rewrote the code published in http://mathworld.wolfram.com/VennDiagram.html to work with the latest version of Mathematica.

VennDiagram.m

VennDiagram[n_, ineqs_: {}] := Module[{i, r = .6, R = 1, v, x, y, f},
  v = Table[Circle[r {Cos[#], Sin[#]} &[2 Pi (i - 1)/n], R], {i, n}];
  {
   If[ineqs == {}, {},
    f = And @@ (v /. Circle[{xx_, yy_}, rr_] :> (x - xx)^2 + (y - yy)^2 < rr^2)[[ineqs]];
    RegionPlot[ImplicitRegion[f,
       {x, y}], Axes -> False,
      DisplayFunction -> Identity][[1]]
    ],
   v
   }
  ]

Show[GraphicsGrid[
  Partition[
   Graphics[VennDiagram[2, #], PlotRange -> All, 
      AspectRatio -> Automatic, 
      PlotLabel -> TraditionalForm[And @@ (Subscript[A, #] & /@ #)] /. 
       True -> \[EmptySet]] & /@ Sort@Subsets[Range[2]], 2]
  ]]

Show[GraphicsGrid[
  Partition[
   Graphics[VennDiagram[3, #], PlotRange -> All, 
      AspectRatio -> Automatic, 
      PlotLabel -> 
        TraditionalForm[And @@ (Subscript[A, #] & /@ #)] /. 
       True -> \[EmptySet]] & /@ Sort@Subsets[Range[3]], 4]
  ]]